The Stability of Canadian Demand for Money Functions 1954-75

The Stability of Canadian Demand for Money Functions 1954-75Author(s): Norman CameronSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 12, No. 2(May, 1979), pp. 258-281Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/134600 .

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The stability of Canadian demand for money functions 1954-75 N O R M A N C A M E R O N / University of Manitoba

Abstract. Stability tests are conducted on Canadian quarterly data for the period 1954-75 for twelve different specifications of fairly traditional demand for money functions. The tests are all based on series of recursive one-period prediction residuals; they test for the presence of serial correlation, heteroscedasticity, misspecification of form, and structural change - either gradual or abrupt. Some problems are encoun- tered in dealing with several types of instability at once. The test results suggest mild heteroscedasticity, strong serial correlation in the untransformed equations, and structural change of some sort for both definitions of money. The 1967 Bank Act revision does not seem to have caused an abrupt shift of demand for either broad or narrow money.

La stablitW desfonctions canadiennes de demande de monnaie 1954-75. L'auteur teste la stabilit6 de douze sp6cifications de fonctions de demande de monnaie en se servant de donn6es trimestrielles pour le Canada entre 1954 et 1975. II s'agit de tests sur des s6ries d'erreurs de pr6diction a partir d'un modele r6cursif de premier ordre. Ces tests d6tectent la presence de corr6lation s6rielle, d'het6rosk6dasticit6, de mauvaises specifications dans la forme du mod&le et de changements structuraux - soit graduel soit abrupt. L'auteur examine les difficult6s rencontr6es dans les cas oil diff6rents types d'instabilit6 coexistent. Les tests suggerent un peu d'het6rosk6dasticit6, une forte corr6lation serielle dans les 6quations non-transform6es, et des changements structuraux pour les deux d6finitions de monnaie utilis6es. II semble que la revision de 1967 dans la Loi sur les Banques n'a pas engendr6 un changement brusque dans la demande de monnaie dans l'une ou l'autre de ses d6finitions.

IN T ROD U CT ION

The demand for money is still at the core of our theory of stabilization policy, despite apparently dramatic about-faces in the behaviour of central banks (see e.g. Courchene, 1976, 24-33; Kane, 1974, 837-9). The stability of that demand for money is rapidly becoming the most critical feature for central bank

Research for this paper has been supported by the Canada Council. The author wishes to thank two anonymous referees, W.C. Riddell, James Mackinnon, C.M. Beach, and John Rowse for helpful comments on earlier drafts. All remaining sins are the author's alone.

Canadian Journal of Economics / Revue canadienne d'Economique, XII, no. 2 May / mai 1979. Printed in Canada / Imprime au Canada.

0008-4085 /79 / 0000-0258 $01.50 / ? 1979 Canadian Economics Association



The stability of Canadian demandfor money functions / 259

policy. If the central bank relies on control of monetary aggregates as its policy instrument, it must believe in a known and reliable connection between changes in that aggregate and changes in the arguments of the money demand function in order for its policy to have predictable effects on those arguments. If instead the central bank relies on interest rates as targets and adjusts the monetary aggregate through daily reserve management to whatever level is required to hit them, instability of the demand for money could make the required reserve changes both large and unpredictable. Disorderly financial markets might well result.

There exists a broad consensus on the nature of the demand for money function, summarized in Laidler (1977, 120-2) and tested comprehensively for Canada by Clinton (1973) and White (1976, section 5). There is less consensus about its stability. Laidler (1977, 133-4) found little variation in interest elasticities over subperiods in work done by himself and others for the United States and the United Kingdom. Khan's (1974) results using recursive residuals formally confirm this result for all parameters collectively for the United States. For Canada, Clinton (1973) found evidence of instability in Chow tests over the period 1955-71, except for the narrowest of his definitions of money. Foot's (1977) revision and extension of Clinton's work basi- cally confirms the latter's main findings but with smaller values of the relevant test statistics. Clark (1973) found broad money to have a stable demand function when tested with annual data for the periods 1927-40 and 1946-65. White (1976, 23, 31) found that all his definitions of money had formally unstable demand functions when he applied Chow tests to a series of forecast periods; the results were the same for quarterly data from 1959 as for monthly data from 1968. Rausser and Laumas (1976, 377) found that the coefficients of the narrow money demand function display systematic variation over the period 1954-71, while those for broad money display only random variation. Finally, Short and Villanueva (1977) found with recursive residuals and Chow tests that the pattern of substitutability had been changing for mrost of the near-money substitutes for currency and demand deposits, suggesting instability in broad money demand functions.

This lack of consensus is not too surprising, since between the studies mentioned there are differences in time periods and frequency of data, in the variables and specifications tested, in the definitions of money, and even in the meaning of the null hypothesis of stability. This irregularity is unfortunate, since much change has taken place in the Canadian financial system since the early 1960s, especially in the money market, and more change will be caused by the adoption of electronic fuinds transfer systems (see. e.g. Binhammer and Williams, 1976). It would be a great help for the future if we had a clearer picture of the impacts of past changes on the market for money.

This paper is an attempt in that direction, using a variety of tests based on the recursive residual approach developed by Brown, Durbin, and Evans



260 / Norman Cameron

(1975) and extended by Riddell (1977).' This approach is applied to twelve different specifications of the demand for money, chosen with relatively few a priori restrictions on the parameters. Nothing like the full range of possible forms is tested, but enough to provide a useful amount of sensitivity analysis.

The paper is arranged as follows. The next section introduces recursive residuals and various test statistics based on them, lays out the full null hypothesis of stability, and then outlines the pattern of test results to be expected from various possible alternatives to that null hypothesis. The third section describes the equations on which stability tests will be conducted. The fourth section presents the actual pattern of test results and interprets that evidence in the light of the discussion in the second section. The final section provides some brief comparison with the previous work reported above, which is followed by a summary and a conclusion.

STABILITY ANALYSIS WITH RECURSIVE RESIDUALS

It has become common to test equations and whole models by looking at their prediction errors (e.g. White, 1976, 42; Selody, 1978, 598), and recursive residuals are merely a series of standardized prediction errors.2 Consider the general model

y = XT,B + U, (1)

where y = (YI, Y2, ... YT)' is a (T x 1) vector of observations on the endogenous variable, XT = (X1, X2, ... XT) is a (T x K) matrix of observations on the K explanatory variables, 1B = ( 13, /32, ... ,3K)' is a (K x 1) vector of unknown coefficients, and u = (ul, u2, ... UT) is a (T x 1) vector of unknown residuals for the periods 1 to T. Let ,B(t - 1) be an estimate of,B based on the series of observations for periods 1 through t - 1. The prediction error for period t is therefore et = Yt - xtf3(t - 1). The recursive residual uft based on forward forecast errors is then3

uft = etl(1 + Xt(X't_IXt_,)-1Xt')"12, (2)

where Xt- I contains only the first t - 1 rows of XT. For a set of time series observations, there will be a whole (T - K) x 1

1 Two of the formal stability tests proposed by Brown, Durbin, and Evans (1975) have been rieported by Short and Villanueva (1977) and by Khan (1974) for the United States and most recently by Pagan (1977) for the Australian demand for money. These studies do not realize much of the potential of the approach, however; Riddell's (1977) application goes much further.

2 This section provides only a brief introduction to recursive residuals. For more rigorous discussion and for proofs the reader is referred to Brown, Durbin, and Evans (1975), Khan (1974), and Riddell (1977), the sources on which this introduction is based.

3 For the reader's convenience, the lettersf and b have been added whenever the variable or statistic symbolized is based on either forward or backward forecast errors respectively. References to test statistics which omit the suffix refer to both versions of the statistic collectively.




vector of these recursive residuals uf = (ufK+l, ... uft ... ufT). One may run the same procedure backwards and generate another vector of recursive residuals, ub = (ubT-K- 1, *-- ubt ... ub,).

These two sets of (T - K) residuals have a number of desirable properties which this paper exploits. First, they are easy to compute in conjunction with algorithms which update parameter estimates to take account of new information. Second, a by-product of the procedure is two sets of (T - K) parameter estimates, starting at either end of the observation range. Third, the residuals are easy to understand intuitively as a form of prediction error; if an equation is underpredicting badly there will be a series of large positive prediction errors and therefore a series of large positive values in uf or ub or both. Fourth, they have the same desirable properties as Theil's (1965) BLUS

residuals, except that they are not 'best' in Theil's sense: under the assumptions denoted Ho below (the null hypothesis) the vectors uf and ub will have the properties E(uf) = E(ub) = 0, E(uf'uf) = E(ub'ub) = u-2I and will be linear in the dependent variable (Riddell, 1977, 8). As LUS residuals they can be used in place of Theil's BLUS residuals in exact tests for serial correlation, for heteroscedasticity, for misspecification of variables, and for structural breaks in the coefficient vector. In general they appear to be only slightly less powerful in those uses than the BLUS residuals (Harvey and Phillips, 1974; Phillips and Harvey, 1974; Riddell, 1977, 12). The assumptions about the true underlying model which make up the null hypothesis are as follows:

XT is non-stochastic and of rank K, u - N(O, E(U'U)),

Ho: E(ut, uS) =z2, if t = s, (3) =0, ift7 s,

/ is constant for all t = 1, ... T,

P(t) is estimated by ordinary least squares.

The fifth desirable feature of recursive residuals is that they react quite distinctly and sometimes differently to various departures from Ho; this property is enhanced by plotting not the residuals themselves but two trans- forms of them, the cusums cu and cusums of squares CUSQ. The forward versions are defined as4

cuf(t) (1/a) Eufj (4)

where - = (I7(uf - uf)2I(T -K_))"2 and t / T

CUSQf(t) = E ufi2 / ufi2. (5)

4 Unless otherwise specified, all summations of recursive residuals start at the beginning of the vector, i.e. ufK?l or ubT-K-1 as appropriate. The particular definition of & in (4) is due to Harvey (in Brown, Durbin, and Evans, 1975, 179); it inflates the cusums somewhat from the original definition by Brown, Durbin, and Evans, increasing their power of test.




The backward cusums cub(t) and cusums of squares cusQb(t) are defined similarly, summing from T - K - 1 backwards to period t.

Tests based on recursive residuals These four series may be merely inspected, along with the parallel series of coefficient vectors /3f(t) and /3b(t), for what they suggest about the nature and timing of departures fi-om Ho. What they might suggest, and how, will be discussed further below. Since the series do have approximately known distributions under Ho, they can also be used together with other test statistics based on recursive residuals for exact formal significance tests of Ho against the various possible alternatives denoted Hj.

The cusum series have under Ho a zero expectation and, approximately,

Var(cuf(t)) = t - K, Var(cub(t)) =T - K -t, cov(cuf(t), cuf(s)) = min(t, s) - K, cov(cub(t), cub(s)) = min(T - t, T - s) - K.

Brown, Durbin, and Evans (1975, 153) have used this information to construct approximate linear confidence bounds on either side of the mean value line for a formal confidence interval test of the actual cuf or cub series.5 UnderHo the cusums of squares series have a beta distribution around an expected path of (t - K)/(T - K) for cusqf and (T - K - t)I(T - K) for cusQb. Approximate confidence bounds have been derived which run parallel to that mean-value line and provide a formal confidence interval test for the actual cusof and cusQb series.6

Other formal test statistics based on recursive residuals are the modified von Neumann ratio Q' (Phillips and Harvey, 1974), the heteroscedasticity ratio HET (Harvey and Phillips, 1974), the Chow F-statistic FCHOW (Riddell, 1977; Chow, 1960), and the Harvey-Collier t-statistic HCt for misspecification of a variable (Harvey and Collier, 1977). Their definitions and distributions are listed below; their significance is apparent from Table 1. Where both forward and backward versions are possible, only the forward version based on uf is defined here.

T-1 T

Qf = t(uft+1-uft)2 Y Uft2 (T - K)/(T - K - 1) (6)

- N (2, 4/( T - K)) approximately when (T - K) > 60 under Ho,

5 The true bounds would be concave but are mathematically intractable (Durbin, 1971). The linear bounds have significance a at the mid-point of the cuf or cub series, and less than that at all other points, a fact that further weakens this test (Brown, Durbin, and Evans 1975, 153).

6 The approximation lies in using the distribution appropriate to a continuous process. This exaggerates the true significance level of the test and thus weakens the power of this test as well (Riddell, 1977).



W A ~ ~ ~ ~ ~~~~~~~A A

Cl &. o = ;

E G >, C' D C l d Z2^4

O Y t: O ~~~~~~~~~O

-Q c O oWat >E d> o : e i < < $ 8 $ o i U g : =




HETf ( uft2/(t1 - K)) ( E Uft2l(T - ti)) (7)

F(T - t1, t1 - K) under H0,

FCHOW = ((?ufP -A )/K )/(A(T - 2K)), (8)

where

tI tl+1 A= uft2+ Z ubt2,

FCHOW - F(K, T - 2K),

HCtf = uft(T - K)i/2) /2 ( ( Uft-uf)2/(T - K - 1))

t(T - K).

The significance of these various test statistics is best discussed by consid- ering the various possible kinds of departures from Ho. As examples of each kind, made very specific only for purposes of discussion, we shall consider, first, the hypothesis of greater variance after some point t1:

H1I: E(ut2) = -12, t < tI (10) = 0J22, t > t l,

j0-2 < 0J23;

secondly, the hypothesis of (first-order) positive serial correlation:

H2: E(utut+1) = pa-2 , O< p< 1; (11)

thirdly, the hypothesis of an upward shift in the coefficient vector after time t1 such that the model becomes:

H3: Yt=Xt/ + Ut, fort =1,.. .t1,

=xtP* + ut, foir t =t, + 1,9 ... T, (12) Ai*=Ai+Et 9 ,> 0, i=l, ... K;

and, finally,the hypothesis (H4) that the ith variable Xi in X (with associated coefficient A3 < 0) is misspecified in Ho. in that it should appear logarithmically rather than arithmetically. These four general kinds of departure from Ho can of course appear together, and when they do the distribution of the test statistics for one violation of Ho may be affected by the presence of the other violation. Table 1 presents this information in compact form.

Rather more elaboration than that provided in Table 1 is needed for a specially troublesome kind of joint violation of Ho: serial correlation and structural change.7 Serial correlation by itself presents difficulties in that a correlation parameter must be estimated and the estimating equation trans-

7 The author is indebted to Riddell and to an anonymous referee for this point.




formed by it to generate the recursive residuals. The fact that the parameter is only an estimate has some (as yet unknown) effect on the distributional properties of the recursive residual statistics and therefore on the properties of tests based upon them. Their power will be weakened, and any given confidence bounds will have a greater significance level than would otherwise be the case.

If there has also been an unknown amount of structural change, it will be in general impossible to get even an unbiased estimate of the serial correlation parameter. Two possibilities arise. First, any gradual upward shift in the function will be at least partly captured and removed by the transformation. Second, if there is an abrupt shift in the function some serial correlation may remain even after the transformation. In the first case the power of recursive residuals tests to detect structural change is weakened. In the second case the recursive residuals will not have the LUS properties needed for the test statistics based upon them.

The only redeeming feature of the second possibility is that evidence of its appearance itself suggests structural change. To cite Pagan (1977, 4), 'In fact, it is frequently the case that patterns in the autocorrelation function that have no simple interpretation can be traced to parameter instability.' This point will be returned to in discussion of the results.

Two other features of Table 1 deserve mention. First and foremost, rejec- tion of Ho by any one of these test statistics supports more than one alternative hypothesis. One can only narrow down the choice of a successor to Ho by looking at combinations of results, especially of pairs of backward and forward test statistics. Second, there is overlap between the two cu series and the HCt statistics, as well as between the CUSQ series and the two HET statistics. A look at their respective definitions will confirm this. The duplica- tion is worthwhile because the four cu and CUSQ series 'provide very useful visual evidence of possible parameter instability and other departures from the null hypothesis. However, the tests do not appear to have high power against a number of alternative hypotheses and therefore should be supple- mented by other procedures for formal significance testing purposes.'8 In particular, the cu and CUSQ series provide some rare information on the broader question of whether the fixed-parameter approach is likely to have the best forecasting performance or whether a time-varying parameter approach would be better.

THE MONEY DEMAND EQUATIONS

The equations considered in this paper all use quarterly data and are specified in several ways for both narrow money (MI) and broad money (M2). MI

8 Riddell, 1977, 25. See the discussion by Khan, Phillips, Anderson, Harvey, and Quandt in Brown, Durbin, and Evans (1975), as well as Garbade (1977) and Goldfeld and Quandt (1976, chap. 1).




corresponds most closely to our a priori idea of means of payment, while M2 is the aggregate which reserve management by the Bank of Canada actually controls. All specifications are fairly traditional, building on White's (1976) results in particular. The relevant variables are assumed to be real income or wealth, the price level, an own-rate on money, and an opportunity rate on alternative assets. Real wealth is approximated by a distributed lag on real income, and to the extent that nominal opportunity rates include some reflec- tion of expected future inflation, that variable is included as well (Laidler, 1977, 136). The particular statistical series used are described in the notes to Table 2.

The general form of relationship used is the following, where lower case letters refer to variables entering logarithmically:

m = 1o +I1Qi +132Q2 +133Q3 +/34y +/35P +f36(Ropp-Rm) + e. (13)

In this equation m refers to the demand for money, Qi are seasonal dummy variables defined by Sparks's method (see Foot, 1977), y refers to real income, p to the price level, Ropp to the opportunity rate of return on money substitutes, and Rm to the own-rate paid on money itself.

Some constraints on the general form of (13) have been necessary for estimation purposes, though they have been kept to a minimum.9 First, the explanatory variables y, p, and (Ropp - Rm) have been fitted with distributed lags of either four, six, or eight quarters, using Sparks's (1967) version of the Almon polynomical technique and choosing the order of polynomial by a combination of Sparks's method and that of Godfrey and Poskitt (1975) (Dhrymes, 1971, 227-9).10 Second, the long-term price elasticity ,85 is constrained a priori to a value of one, in view of White's finding (1976, 22f) that,85 and /84 could not be simultaneously estimated with any degree of accuracy and Laidler's evidence (1977, 138-9) for the neutrality of money. The use of a distributed lag for this term permits money illusion, but only in the short term. Third, the own-rate on MI is assumed constant, despite Klein's (1974) sug- gestions that the implicit rate varies; its influence is therefore merged with that of the constant term. Fourth, all the equations estimated have been trans- formed to remove the first-order serial correlation in the original error term. The autocorrelation parameter has been estimated for the full sample period using a modified Hildreth-Lu technique (Beach and Mackinnon, 1978).

Table 2 presents the main parameter estimates from a number of variants of

9 Equation (14) already imposes the constraint that a 100-basis point change in the interest rate has the same effect whatever its initial level. This follows White's (1976) approach rather than that of Clinton (1973) or Foot (1977). For the MI equations, this hypothesis is checked with the HCt statistics reported in Table 3 below.

10 The use of a Koyck lag form is prohibited by the requirement under Ho that X be non- stochastic. The use of separate Almon lags for each variable also allows different speeds of adjustment to different variables. All lag forms are constrained to a weight of zero at the end. See Frost (1975) for the significance of errors in specifying lag orders and lengths.




TABLE 2

Regression estimates, full sample period

Interest rate term Real income term Serial correl-

Equa- Defini- Lag Elas- Lag Lag Elas- ation tion tion length ticitya length orderb ticity coeff. RMSE

MI equationsc A RFP 4 -0.115 6 1 0.689 0.84 0.0164 B RFP 4 -0.126 6 3 0.705 0.84 0.0160 C RFP 8 -0.178 6 3 0.748 0.88 0.0161 D RTB 8 -0.163 6 3 0.751 0.79 0.0160 E RFP 4 -0.121 4 2 0.701 0.84 0.0160 F RFP 4 -0.121 6 2 0.698 0.84 0.0163 M2 equationsd G RFP-RM2 4 -0.147 6 3 1.07 0.96 0.0146 H RFP-RM2 4 -0.136 6 2 1.05 0.96 0.0149 I RFP-RM2 8 -0.257 6 2 1.06 0.97 0.0137 J RTB-RM2 8 -0.266 6 2 1.10 0.94 0.0131 K RFP-RM2* 4 -0.152 6 2 1.05 0.96 0.0146 L RFP-RM2 8 -0.257 6 3 1.08 0.96 0.0141

a The elasticity is measured at the average level of the opportunity rate of return. For the M2 equations it is therefore a partial elasticity with respect to the opportunity rate, holding the own-rate constant.

b All third-order polynomial lag forms are also constrained to a slope of zero at the end, so only the last two Almon terms enter the estimating equation. This is the result of the Sparks (1967) procedure for choosing Almon lag forms.

c The price level term enters equations A to F as a four-quarter, first-orderAlmon variable subtracted from the dependent variable.

d The price level term has a six-quarter lag form of order one for equations G and L, of order two for the others.

NOTE: Durbin-Watson statistics are reported below with the other test statistics. All elasticities are highly significant. All R2 are 0.99. Wherever the data permit, the period is 1/54 to 3/75. For equations c, i, and L it begins four quarters later. MI: currency in circulation plus demand deposits in the hands of the public, excluding float, average of Wednesdays in the last month of each quarter, not seasonally adjusted M2: MI plus personal savings deposits (PSD's) and non-personal term and notice deposits (NPTND'S) RFP: 90-day financial paper rate, average of month-end data; mean = 5.24 RTB: 90-day treasury bill rate, average of month-end data; mean = 4.29 RM2: weighted average of rates paid on chequable savings deposits, non-chequable savings deposits, and 90-day deposit receipts; the weights are lagged shares in the total of (PSD + NPTND); the 90-day deposit receipt rate is a proxy for all rates paid on NPTND'S. Average of month-end data; mean = 3.76 RM2*: derived like RM2, but excluding the rate paid on chequable savings deposits after 1967, when non-chequable savings deposits appeared; mean = 4.02. Real income: real gross national expenditure, not seasonally adjusted. The price index used is the real GNE deflator. RMSE: root mean squared error




(13) for the period starting with the first quarter of 1954 (1154) and ending with the third quarter of 1975 (3/75). The period therefore extends from after the end of the bond support program to before the imposition of wage and price controls. For each of MI and M2 a number of different lag form combinations have been used, as well as two interest rates representing the opportunity cost and two representing the own-rate for M2.1' The estimated parameters are quite typical of other results in this area and call for little comment. The high serial correlation coefficients are cause for concern here, as in White's (1976) results. Further tests of Ho are reported and then evaluated in the next section.

STABILITY TEST RESULTS

The test statistics presented in Table 1 have been calculated for each of the equations in Table 2 to follow up the various alternative hypotheses. They are reported in Tables 3 and 4 for the MI and M2 equations respectively. In addition, the plots of cuf, cub and cusQf, cusQb are given in Figures I to 4 for equations A and G. They are typical of the plots for the other MI and M2 equations respectively, as will become clear below. The approximate linear confidence bounds given are for a significance level a of 5 per cent unless otherwise specified; only parts of these boundary lines are drawn in, to reduce congestion.

Some comments are required on how the various test statistics from equations (4) to (9) have been calculated and reported in Tables 3 and 4. Then the results will be compared with the possible response patterns from Table 1.

The cusum and cusum of squares series take up too much space to display in full for twelve equations; yet there is some variation in them between equations. Since the cusum series have approximately the same general shape as Figure 1 (for MI equations) or Figure 2 (for M2 equations), the differences are mainly ones of average slope, which can be fairly well reflected in the reported values of the terminal cusum statistic (i.e. cuf(T) or cub(l)). For the cusum of squares series, the general shape can be described succinctly in terms of concavity or convexity as seen from below. As a guide to interpreting this, CUSQ series which are sharply convex over a long period will generally lie below their mean-value line for most of that period and will approach or cross their lower bound.

The modified von Neumann ratio Q' has not been calculated where the Durbin-Watson statistic lies above the upper bound at 5 per cent, since that is itself a very stringent test. Since the Hildreth-Lu procedure has already been used before estimating these equations, a two-tail test is appropriate for Qf' and Qb'. The HETf and HETb statistics reported use the first K and last K

11 Use of other short-term rates is not possible for this long a period; either the data do not go that far back, or else the rates are clearly not representative of the whole structure of short-term opportunity rates.



-iz V V) 1O 0 00 * ? t i- O C - U N*

u 4 btZ - j 00

cq cq 00 00

? - N t Cl O N --

V)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1 V)~~~~~~~~~~~~~~~~~~~~~~1

c o 0~~~~0

- . . C

- 0 0

- Cl ~~~~~~~~~~o 00 - e~~~~~~~~~Cd l

- Cl Cl - -- C CI

0

- N 4" g o t~1) ? N C Col 00 00 00 ,

x 00 0 m)

1 00 N _, 20 N N

Ut u$ S

=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 0 C's

Z A Ax AA: E~ A A EE

0~~~~~~~~~~~

U > c cq >

.o s_ 2 3

t~~ ~ ~~~~~~~~~~~~ en N N N kt) t

Cld ~ ~ ~ ~ C o ,U.ottt

m~~~~~~~~~~~~~~~~~~ ur)* OO



J:- - m =b

c r) oo ?o q N 7: "o 00 ? =o m m .0

=U ~~~~~~~~~~~N 00 W)Nm

W N

00 IN . o

4 ? ̂ ca~~~~~~Cd t c at

:~~~~~~~~~~~~U _) r

U >

? X 0 0 cA, Ll Ll Ll :

.o > a~

*Ct o Z

A~~~~~~~~~~~~~~~i ci 'A cis *A Lt( *d(



The stability of Canadian demandfor moneyfunctions I 271

25- CUfMOx n.05 a=.05 CUma

20 - ox

15CU

I0

-5 55 60 65 70 75

FIGURE 1 Cumulative sum of standardized prediction residuals: cuf, cub for Ml equation A (a is two-tail significance level of approximate linear bound)

30 CUf m ax

25-

20 a= ? ;

a=.05 15 -

CUb max I0

a= .05 5

CUf Cub 0-

-5

-10 CUb m i II l L IS_L ,1 1 1 1 1 1 1, I I Ii I 1

55 60 65 70 75

FIGURE 2 Cumulative sum of standardized prediction residuals: cuf, cub for M2, equationG (a is two-tail significance level of approximate linear bound)




CUSQf / . O. f max -0.9

02- 0.8 0.3 ~~~~~~~ ~ ~~~CUSQb ' USQf 0.3 - JI~~~~~(Lfcae (right scale) -0.7 030 ~~~~~~~~~~~~(left _,Jscole) 00

0.4 _ - - _ 0.6

0.5- - 0.5

0.6- 0.4 CUS xb CUS/ /U Q min ?

0.7- max -~~~~~~~~~~~~~~~0.3

0.8 - ,---' , o _ 0.2

0).9- ~ ,'CUSQb min -0.1

1.0 ,$S 1 t I 1/ t 1 1 I I, R I I . I l I I I | l I I I | )0 55 60 665 70 75

FIGURE 3 Cumulative sum of squares ratio: cusQf, cusQb for Ml, equation A

0 - 1.0

0.1 CUSQf 0.9

0.2 ,USQb -0.8 / scale) CUSQf

0.3 - ~~~~~~~~~~~~~~(rgtscale) -0.7

0.4- - 0.6

0.5 - _0.5 CUSQf min

0.6 CSb-Q4

0.1~~~~~~~~~~~~~~~~~~. 0.8 v 011-41 Q2

0.9~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. 0CUSQb min

I I ii I 0 - l 55 60 65 70 75

FIGURE 4 Cumulative sum of squares ratio: cusqf, cusQb for M2, equation G

observations respectively as their basis, which does not yield the most powerful test (Harvey and Phillips, 1974). The mid-point for the test (t1) was found by inspection of the cusqf series for each equation (see Figures 3 and 4). For the MI equations the cusqf series all rise relatively steeply after 4/71 and relatively slowly after 4/62, which suggests a partition of the observation period into either 1/54 to 4/71 and 1/72 to 3/75 or the same starting 1/63. For the M2 equations the cusqf series rise relatively slowly from 3/55 to about 2/67 and steeply thereafter, so only that partitioning is used.




The FCHOW statistic requires an advance choice of switching points and of the number of coefficient regimes to be considered. One of the more desirable features of recursive residuals is that they are affected by structural change only when it happens, so the date of a (first) structural change can be inferred as the point at which the cuf series starts to veer away from zero12 If there are three regimes, the second switching point will be hard to detect from the cuf series since its expected value is unknown a priori after the first switching point. However, it should be the point at which the cub series veers away from zero. The cuf and cub series together will therefore provide evidence not only on the timing of structural changes but also on whether there are two or three changes. 13

What emerges clearly and uniformly from the cuf series for MI (Figure 1) is an upward kink at around 1/65, generally after a more or less steady but slow upward drift to that point. The cub series for MI are random back to around 2/69, and trend upward throughout the 1960s. This change is less pronounced and uniform across MI equations than the kink at 1/65, so that two alternative hypotheses are tested with FCHOW - one with a single break after 1/65 and another with a second break after 2/69. For the M2 equations (Figure 2) the upward kink in cuf is just as strong and uniform but about two quarters earlier, at 3/64. The cub series are less uniform but show a downward trend only before 2/70, so the same strategy is used here too.

For the maximum power of the Harvey-Collier t-statistic, the observations should be arranged in advance in the same order as the values of the variable whose specification is suspect. In these equations the most plausible alternative specification involves entering the interest rate terms logarithmically, so that for maximum power the observations should be ordered by the size of whichever interest rate term is being used in the equation. This is impossible in time series work with distributed lag formulations, because the observations must be in historical sequence. However, the RTB and RFP series used in the MI equations do rise more or less evenly over the sample period, so the HCt statistic does have meaning and some power and is reported. For the M2 equations the interest rate differential used is not remotely parallel to the time sequence, so this test statistic would be meaningless and is not reported.

Evaluation of MI results Table 3 does not conform exactly to any of the response patterns of Table 1, suggesting more than one kind of departure from Ho. Each of the alternative hypotheses H1 to H4 will be considered in turn.

12 Short and Villanueva (1977, 443) use the cusQf series to detect likely dates for structural change. This seems less reliable than using cuf and cub, since the CUSQ series would not reflect a long series of underpredictions unless they were larger than average. Of course, the date or dates on which either a cu or a cusQ series actually crosses its confidence bound are meaningless; only the dates of kinks in'the series are useful.

13 If there are more than three regimes, a variable-parameter regression approach may be more useful (see Garbade 1977).




The increasing variance hypothesis HI is supported and Ho is often rejected by the HET statistics which start at 1963. The HET values based on the full period are smaller because the years 1956-63 (or 1954-8 for the backward forecasts) have greater variances than the years 1963-71, but even those HET

values are all greater than one. HI is supported in part by the pattern of the CUSQ series; both series rise relatively slowly over the years 1963-71 and relatively quickly thereafter, confirming the HET results for those two periods.

Accepting HI for the two periods 1963-71 and 1971-5 has implications for the distributional properties of the other test statistics (see Table 1). Toyoda (1974) has shown that it biases FCHOW upwards, roughly doubling the true probability of the 5 per cent limit under Ho. It twists the mean-value line and therefore the confidence bands of the CUSQ series, and it twists the exact confidence bands for the cu series as well. For the years 1963-71 the cu bands should draw closer; for 1971-5 they should spread. In this case it merely corrects some of the inaccuracy of the linear confidence bands used, which under Ho are relatively too close for the middle years and too far out for the end years.

The hypothesis of first-order autocorrelation H2 is only very weakly supported, and Ho is not rejected for any of the specifications tested. This suggests there is no more serial correlation than the considerable amount already removed and reported in Table 2. The reader should be aware of two possibilities raised earlier: that the estimated serial correlation parameter may have been biased upward by the presence of gradual structural change, and that the presence of an abrupt structural shift might cause the transformation for serial correlation to fail to remove that serial correlation. For these equations the latter does not seem to have happened. The former may have; if so, the cusum tests are weakened and are less likely to detect structural change.

Nevertheless, the hypothesis of structural change H3 is supported with varying strength by several statistics. Ho is rejected for three specifications and nearly rejected for the others by the CUF series. There is a recognizable point at which the cuf series veer off, though really they only veer off more quickly after that point. The forward HCt statistic, which is a transformation of the cu(T) value, rejects Ho in all six specifications. The pattern of the cu series support H3 in that the largest errors in both series occur towards the end of the forecast series uf and ub, causing cusQf to be convex and cusQb to be concave.

On the other side, the FCHOW statistics do not come close to rejecting Ho in favour of either the two-regime or, especially, the three-regime hypothesis, despite the high power this kind of two-stage test should have. This may reveal that the proper form of structural change hypothesis involves, not switching fixed coefficient regimes, but rather time-varying parameters of some sort.

The hypothesis of specification error for the interest rate term H4 is also well supported. The HCt statistics reject Ho forwards, though not backwards;




the fact that the cu series wander off quite early (though slowly till 1/65) supports H4 rather than the particular form of H3 proposed. The pattern of the CUSQ series supports H4 in the same way that it supports H3. The behaviour of the cu over the period 1970-2 also supports rather strongly the idea that RFP

and RTB should enter the equation logarithmically: when the interest rate dropped abruptly in 1970 to the levels of 1964, the forecast errors uf and ub fell; when the rate rose precipitously in 1973 the cu series picked up again.

On the other side, the backward HCt statistics, except that for equation B, do not support H4, let alone reject Ho. To test this further, equation F was changed by entering RFP logarithmically. The equation's performance was not improved: the HCt statistics went up; the DW statistic went down; HETf went down and HETb up; and both FCHOW statistics rose. One can conclude that at least that form of misspecification is not the problem, and if it is the only one that suggests itself some form of structural change, together with limited heteroscedasticity, becomes the best substitute for Ho for the MI equations.

Evaluation of M2 results For the M2 equations, the heteroscedasticity hypothesis H1 is supported and Ho is rejected by the HETf and cusQf statistics. The HETb statistics do not reject Ho, and the cusQb series suggest only that the variances for 1962-7 are smaller and those for 1967--70 larger than normal. The errors for 1970-4 and for 1954-60 are larger only in cusqf and cusQb respectively, i.e. when they are at the end of the series of forecasts, a fact that does not support H1. Acceptance of HI for the middle period will change the distributional properties of the other statistics, as was discussed above for MI, but the degree of heteroscedasticity does not appear to be large.

The autocorrelation hypothesis H2 is supported strongly by the Durbin- Watson statistics and by the more exact and powerful Q' statistics; for four of the equations Qf' rejects Ho in a two-tail test, and on the side indicating positive correlation. This strong evidence of positive serial correlation of uf raises the serious econometric problem alluded to earlier in the paper. If there has been a large structural break in the model, the Hildreth-Lu procedure applied to the full sample period will still leave serial correlation in the u vector, and the distributional properties of the uf and ub series will be altered dramatically. In a simple case, Cox showed that first-order serial correlation would inflate the cuf series by a factor of two if the coefficient of correlation were as large as 0.6 (Brown, Durbin, and Evans, 1975, 164; see also ibid., 182, 184, 189). None of the cuf series in Table 4 exceed their limits by that much. The confidence limits of the CUSQ series would be wider as well under those circumstances, but since that is expressed in ratio form the expansion would probably not be as large.

One cannot deny these possibilities, but for this case there are two reasons not to be pessimistic. First, some of the results are not consistent with the idea of a second round of positive serial correlation in the original e series of equation (13) after the transformation reported in Table 2. The estimated




correlation coefficients are large (0.94 to 0.97), very close to their a priori maximum of 1.0. There just is not room for further positive serial correlation that, because of structural change, might have gone undetected by the Hildreth-Lu procedure. Second, another explanation of much of the behaviour of the ufand ub series is better supported. If there is a structural break in the underlying model, one of the consequences will be serial correlation of the uf series after the break and of the ub series before the break. These will cause the Q' statistics for the full sample period to fall below the expected value of 2.0, as has been observed. Another consequence which can be used as a check is that the structural break will not cause serial correlation in the remaining parts of ufand ub. If one knows the location of any structural break, the Q' statistics can be calculated for those subperiods to test the hypothesis of no serial correlation in that part of the vector. Table 5 reports the results of such a test, using as break-points the dates from the successful (H0-rejecting) three-regime FCHOW tests reported in Table 4. The Qf' statistics are all higher, generally much higher, than in Table 4, and all but one of the twelve statistics support Ho.

This does not establish that u is free from serial correlation. Indeed, given the evidence of a structural break it is hard to believe the serial correlation parameter has been estimated without some bias. What the results do suggest, however, is that the amount of positive serial correlation left in the u vector independent of structural change is likely to be fairly small. This would make the already approximate boundary statistics too stringent, but by rather small amounts.

The hypothesis of structural change H3 is well supported by the cuf series, by the pattern at the ends of the CUSQ series (and by difference between the HETf and HETb values), and by the FCHOW statistics, which reject Ho for five equations in favour of either a two-regime alternative or a three-regime model, changing after 3/64 and after 1/70. H3 is also well supported by the pattern of Qf' and Qb' results in Tables 4 and 5, as has been mentioned above. On the other side, the cub series are more or less random back to 1969 and do not begin to reject Ho; their abrupt drop in 1970-69 is consistent with a break in the model at that point, but no more. Further, the possible presence of serial correlation might account for some of the high test-statistic values in Table 4.

The hypothesis of misspecification of one of the variables H4 has not been tested for these equations. The suspect variable is the interest differential, but since there is not even a discernible monotonic relationship between that and the order of the observations, the HCt statistic would not reflect misspecification of that term. It would instead reflect H3, and since it is merely a transform of the final cusum value it would have been high for all these equations.

Estimated coefficient changes Since the alternative hypothesis of structural shifts of some type is supported for both sets of equations, it is worthwhile to complete this section with a look



The stability of Canadian demandfor money functions I 277

TABLE 5 Von Neumann ratios for subperiods

QfI Qb' Equation 1/54 to 3/64 3/75 to 2/70

G 2.07 1.84 H 1.73 2.67 I 1.81 1.42 J 1.49 1.90 K 1.73 2.61 L 1.86 1.37

NOTE: The critical levels of Qf' at 10 per cent for a two-tail test are 1.50 and 2.62. At 2 per cent they are 1.29 and 2.83. At 5 per cent they are not available. For Qb', with only 14 d.f., the 10 per cent limits are 1.29 and 2.99, while the 2 per cent limits are 0.96 and 3.35.

TABLE 6

Average parameter estimates for subperiods

Average Income Interest rate Interest income

Period Constant elasticity coefficient elasticity lag

MI equations (A to F) Up to 1/65 1.21 0.58 a -0.036a -0.13 1.37 2/65 to 3/75 3.88a 0.86a - 0.026a -0.16 1.84 M2 equations (G, H, I, K, L)

Up to 3/64 -1.37b 0.69a - 0.041 a -0.046 2.03 4/64 to 1/70 - 7.91 a 1.40a -0.039b -0.080 0.78 2/70 to 3/75 7.75b _0.16b 0.017b 0.021 1.68

a Significantly different from zero at the 5 per cent level in all equations making up the average (not applicable to the last two columns). b Not significantly different from zero in more than two of the equations

making up the average. NOTE: For M2 equations, interest elasticity is the coefficient multiplied by the average interest differential over the subperiod.

at the specific kinds of shifts revealed by the data. The estimated changes in the error variance are already revealed by the HET statistlics, which are ratios of estimated variances. Changes in most of the other parameter estimates are given in Table 6.14

14 There is little change in the seasonal coefficients. The constant term is the 'normal' intercept discussed by Foot (1977), independent of seasonal factors. For equations H, I, and K, in which a second-order Almon lag form is used for the price level, the average lag for the price level variable also changes. It rises from the first period to the second and then returns to more or less its original value. This is just opposite to the changes in average income lags for all of the M2 equations.




The demand for MI appears to have become more sensitive to income (but less quickly) and less sensitive to a 100-basis point change in the opportunity rate, while shifting upwards substantially.15 The demand for M2 is much harder to follow; many of the estimates not only are not precise in the individual equations but also vary widely between equations for the last subperiod. The appropriate lesson to draw from the results for the latest period is probably that there are some misspecifications of asset holders' decision processes16: perhaps they respond to different interest rates; perhaps their expectations-formation process changed after 1969; perhaps real income is not an adequate proxy for real wealth in the 1970s, at least with the lag forms used here. The suggestion of a much faster and much more elastic response of M2 to income in the late sixties is common to all five of the equations, though, and does not seem to be affected by whether the average price term lag is allowed to become much longer in this period (see n. 15). The subsequent slowdown is not common to all five equations. The substantial fall in the intercept term between the first two subperiods is wholly accounted for by the doubled value of the income elasticity (again, see n. 15).

The MI result provides a reassuring answer to the question first raised by Gurley and Shaw (1956): will the introduction of closer substitutes for MI cause increasing interest rate sensitivity of MI, and thereby 'weaken' the ability of the central bank to influence interest rates without excessively large open market operations? (See Feige and Pearce, 1977, for a summary of other answers.) The two subperiods up to and after 1/65 differ particularly in the greater availability of a wide range of MI substitutes in the later period. Non-chequable personal savings deposits are merely one example. The estimated interest sensitivity has declined by over 25 per cent; the implied elasticity has risen by less than that; and apart from considerations of instability there is no prospect of an end to the central bank's ability to control MI through interest rates. 17

CONCLUSIONS AND COMPARISONS

The substantive conclusions of this paper about the demand for money are briefly the following. The null hypothesis of the classical regression model does not stand up well for either MI or M2 demand. There is pronounced serial correlation in both, some of which can be removed from the estimating

15 The effect of a rise in income elasticity, together with a rise in income level, would normally be to cause the estimated intercept term to fall. The same effect would occur with a fall in interest sensitivity and a rise in interest rates using this specification, so the rise in the estimated intercept term is all the more remarkable.

16 White's results (1976, 46ff) are even more pessimistic in that he found instability of some sort for both of the money definitions of this paper over the short period 11/68 to 9/74.

17 If one believes that the higher nominal rates of the later period reflect mainly a higher compensatory margin for greater expected inflation, and if one is mainly interested in an elasticity of demand for MI with respect to the real interest rate, then the rise in implied elasticities in Table 6 is bogus, and a better measure of structural change is the change in interest coefficients.




equation by the Hildreth-Lu procedure but remains to be explained in a fully successful specification. There is mild heteroscedasticity in the form of lower variance after 1963 for both MI and M2, and higher variance for the years 1971-5 (for MI) and 1967-70 (for M2). The underlying residuals are serially independent for the MI equations and seem to be so for the M2 equations as well, despite clear evidence of serial correlation of the prediction residuals. The latter does not appear until after the point at which the cusum series veer away from zero and therefore supports the idea of structural change at those points instead. Evidence of structural change is found in both sets of equations, though less certainly for MI where Chow tests which would pick up a single change of coefficient regime fairly well fail to record one. Evidence for misspecification of the interest rate term for MI is strangely inconclusive. Finally, the parameter estimates for subperiods suggested by the residuals plots are indeed substantially different. The M2 parameter estimates vary widely across equations and are not precise within equations for the 1970s; further work is in order here for those interested in the consequences of the central bank's reserve management operations, since it is M2 which those reserves support.

These conclusions are only as robust as the range of specifications tested, and those are all fairly traditional. At the same time, relatively few a priori restrictions have been put on the specifications, particularly on the lag forms, so that if stable traditional forms existed they would have a good chance of 'emerging' from the data. Also, White (1976) and Clinton (1973) have between them tested a vast array of alternative simple forms and variables, without coming to the conclusion that the general kind of simple function tested here is clearly inferior.

The results presented in this paper do not differ much from those of other studies in this area. Narrow money proves generally more stable than broad money, which generally fails formal stability tests over subperiods since the 1950s (White, 1976; Clinton, 1973; Foot, 1977). This is in line with Short and Villanueva's (1977) finding that demand functions for the whole range of MI-substitute assets are not stable over periods up to 1973. Not surprisingly, the pattern of residuals for MI and M2 is broadly similar to that produced by White's (1976, 23, 42) series of six-month forecast errors in his estimations with monthly data starting in 1959.

In contrast, the a priori assumptions by Clinton and Foot of structural breaks at the points of shift to a fixed exchange rate regime (2/62) and revision to the Bank Act (2/67) are not supported by the series of forecast residuals of this paper. The cuf series for both MI and M2 do not veer noticeably away from zero after 2/62, and there is no perceptible change in its trend after 2/67, as one would expect after a structural break at either date.'8 Foot's assump-

18 This is not to say that a Chow test assuming a break at that point would fail to reject Ho. In fact, for the longer period considered in this paper, Chow tests applied at that point for equations A and G both reject Ho. The cuf series merely suggests that Clinton and Foot have chosen the wrong alternative hypothesis to replace Ho.




tion of a structural break at the point of return to an effective floating-rate regime, dated approximately at 1/71, is much better supported by the behaviour of the cuf series for MI, which veer sharply upwards after that point. The cub series, which should pick this up as well, do not reflect anything dramatic near 1/71 for MI, but suggest a break for M2 only one year earlier. Clearly, Foot's date cannot be rejected.

Rausser and Laumas's (1976) results may also be consistent with the conclusions of this paper. Their stability test in a random-coefficients model essentially measures the ratio of (persistent) variance in the 13 vector to (transitory) variance in a form of disturbance term. It is easy to imagine that more or less continual but gradual change in the /3 vector in the MI equation could cause a relatively large value of that ratio. A pattern of larger but fewer and less continuous shifts in the:/ vector for the M2 equations might cause more of the variation to appear transitory. The results of Tables 3 and 4 are consistent with such patterns of structural change.

The procedural conclusion of this paper is that the recursive residual approach to stability analysis is well worthwhile. The forward and backward forecast residuals are of direct use in estimating the timing of structural changes, and with little extra computation they can be used to proSluce several (whole series of) complementary test statistics of quite reasonable power. More important than the power of each individual test-statistic for a particular type of departure from the general null hypothesis is the ability of the com- bined series of test-statistics to discriminate between alternative hypotheses. This paper provides several examples. Since the null hypothesis of stability has several possible alternatives with quite different implications, this ability is really quite useful. The simultaneous occurrence of serial correlation and structural change presents abnormally severe difficulties and in general weakens the power and validity of the tests used. The approach is capable of shedding light even on that problem, however, and should be rejected only when a better alternative is put forward.

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The Stability of Canadian Demand for Money Functions 1954-75

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